1. Field of the Invention
The present invention relates to a calculation method and an interferometer which calculates the geometrical distance of a test optical path on the basis of the optical path length of the test optical path.
2. Description of the Related Art
A well-known problem in the field of interferometric metrology involves the accurate measurement of distances. Hitherto, a distance measurement method has been well known which is based on interferometric measurement. In its simplest form, an interferometric measurement includes interfering a reference beam with a test beam and obtaining an optical path difference (OPD) between the two beams by analyzing an interference pattern (fringes). The fringes are formed by interfering the reference beam reflected by a reference surface functioning as a length-measurement reference and the test beam reflected by a test surface attached to an object whose distance or surface profile is desired to be measured. Calculation of the OPD is generally based on the phase and wavelength of the interfering beams. However, since the refractive index in the air varies in accordance with the temperature, pressure, humidity and so on, the wavelength on which the measurement is based may vary. Accordingly, the distance measurement in the air may require correction of the refractive-index with high precision.
Two refractive-index correction methods are roughly available. One method measures a refractive index at one point and applies the refractive index measured value to the entire test optical path. This method is generally well used but assumes that the refractive index is even between the refractive index measurement point and the test optical path. For high-precision refractive index correction, a high-precision air conditioner may be required. Another method uses a refractive index dispersion in which a refractive index depends on the wavelength of light to measure the refractive index and the distance at the same time. This method is known as the Two-Color method because two or more wavelengths are used to measure an optical path length. The Two-Color method allows measurement of an average refractive index of a test optical path and reduces the influence of the refractive index dispersion in the air and allows measurement of a refractive index with high precision. However, a problem exists that the measurement resolution decreases by the variance ratio of a refractive index called A coefficient. A method using a moving average has been known in order to solve the problem of the Two-Color method. According to the method described in Japanese Patent Publication No. 7-81819, the moving average in Expression (1) below is used to calculate a length measurement value L to improve the resolution.
                    {                                                            L                =                                                      OPL                    1                                    -                                      A                    ·                                          (                                                                                                    1                            N                                                    ⁢                                                                                    ∑                                                              j                                =                                1                                                            N                                                        ⁢                                                          OPL                                                              1                                ⁢                                                                                                                                  ⁢                                j                                                                                                                                    -                                                                              1                            N                                                    ⁢                                                                                    ∑                                                              j                                =                                1                                                            N                                                        ⁢                                                          OPL                                                              2                                ⁢                                                                                                                                  ⁢                                j                                                                                                                                                        )                                                                                                                                              A                =                                                                            K                      ⁡                                              (                                                  λ                          1                                                )                                                              -                    1                                                                              K                      ⁡                                              (                                                  λ                          1                                                )                                                              -                                          K                      ⁡                                              (                                                  λ                          2                                                )                                                                                                                                                    (        1        )            
Here, OPL1j and OPL2j are the jth optical path lengths measured with a first wavelength λ1 and a second wavelength λ2, respectively, N is the number of times of moving average, and K(λ) is a refractive index dispersion term of dry air.
However, the method described in Japanese Patent Publication No. 7-81819 performs moving average not only on a refractive index but also on a component that depends on a geometrical distance which is the measurement result. For that reason, when an object is driven, the moving average is performed on data including a past driven distance, which delays the response speed and may cause a length measurement error.